The MCMC results in this study successfully capture the true
parameters and their uncertainties. The results contain natural biases arising from the use of prior distributions, internal variability of the climate, and use of a
single noisy sample as synthetic data. Despite the sampling variability and emulator constraints, our
MCMC samples were able to capture the true parameters within an estimated 99% confidence
interval in our examples, demonstrating the potential of
EKI-trained GP emulators for MCMC sampling. Validation of the emulator (Figure REF ) further supports the MCMC results,
as do our comparisons with MCMC using the benchmark emulator
(Table REF ). The GP emulator both smooths the objective function and allows us to quantify uncertainty by sampling from the posterior distribution. This contrasts with uncertainty quantification based on the EKI ensemble, which underestimates the true uncertainties in our experiments by an order of magnitude. As used here, EKI should be viewed as an optimization algorithm and not a sampling algorithm.
Adding additional spread to match the posterior
within EKI may be achieved for
Gaussian posteriors [1]}, [2]} or by means of EKS [3]}; however, these methods
are not justifiable beyond the Gaussian setting. The
MCMC algorithm within CES, on the other hand, samples from an
approximate posterior distribution and is justifiable
beyond the Gaussian posterior setting [4]}.
